A Conserved Cross Helicity for Non-Barotropic

Home , Barotropic fluid

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dν 1 2 A ≡ ∇~ χ · ∇~ χ = ~v − w, (13) ij i j (10) dt 2 ∂ρ and through the specific internal energy ε which is a func- + ∇~ · (ρ~v)=0, (14) ∂t tion of density and entropy. Einstein summation conven- dσ tion is assumed throughout. = T, (15) dt Variational principles for magnetohydrodynamics were dα ∇~ η · J~ introduced by previous authors both in Lagrangian and = , (16) Eulerian form. Sturrock [8] has discussed in his book a dt ρ Lagrangian variational formalism for magnetohydrody- dβ ∇~ χ · J~ namics. Vladimirov and Moffatt [11] in a series of pa- = − . (17) dt ρ pers have discussed an Eulerian variational principle for incompressible magnetohydrodynamics. However, their In the above: w is the specific enthalpy and the current variational principle contained three more functions in ~ ∇×~ B~ is J = 4π . The above equations are shown [9] to be addition to the seven variables which appear in the stan- equivalent to the non barotropic MHD equations (2-6). dard equations of incompressible magnetohydrodynam- The function ν whose material derivative is given in ics which are the magnetic field B~ the velocity field ~v equation (13) can be multiple valued as only its gradient and the pressure P . Kats [12] has generalized Moffatt’s appears in the velocity (12). However, the discontinuity work for compressible non barotropic flows but without of ν is a conserved quantity : reducing the number of functions and the computational d[ν] load. Sakurai [10] has introduced a two function Eule- =0. (18) rian variational principle for force-free magnetohydrody- dt namics and used it as a basis of a numerical scheme, his since the right hand side of equation (13) are physical method is discussed in a book by Sturrock [8]. Yahalom and hence single valued quantities. A similar equation & Lynden-Bell [7] combined the Lagrangian of Sturrock hold also for barotropic fluid dynamics and barotropic [8] with the Lagrangian of Sakurai [10] to obtain an Eule- rian MHD [7, 17–19]. Lagrangian principle for barotropic magnetohydro- Let us now write the cross helicity given in equation dynamics which will depend on only six functions. The (1) in terms of equation (8) and equation (12), this will variational derivative of this Lagrangian produced all the take the form: equations needed to describe barotropic magnetohydrodynamics without any additional constraints. The equa- H = dΦ[ν]+ dΦ σdS (19) tions obtained resembled the equations of Frenkel, Levich C Z Z I & Stilman [13] (see also [14]). Yahalom [18] have shown that for the barotropic case four functions will suffice. in which: dΦ = B~ · dA~ = ∇~ χ × ∇~ η · dA~ = dχ dη and Moreover, it was shown that the cuts of some of those the closed line integral is taken along a magnetic field functions [19] are topological local conserved quantities. line. dΦ is a magnetic flux element which is comoving Previous work was concerned only with barotropic according to equation (2) and dA~ is an infinitesimal area magnetohydrodynamics. Variational principles of non element. Although the cross helicity is not conserved for barotropic magnetohydrodynamics can be found in the non-barotropic flows, looking at the right hand side we work of Bekenstein & Oron [15] in terms of 15 functions see that it is made of a sum of two terms. One which and V.A. Kats [12] in terms of 20 functions. Morrison is conserved as both dΦ and [ν] are comoving and one [16] has suggested a Hamiltonian approach but this also which is not. This suggests the following definition for depends on 8 canonical variables (see table 2 [16]). The the non barotropic cross helicity HCNB: variational principle introduced in [9] show that only five functions will suffice to describe non barotropic MHD in HCNB ≡ dΦ[ν]= HC − dΦ .σdS (20) the case that we enforce a Sakurai [10] representation for Z Z I the magnetic field. Which can be written in a more conventional form: The variational equations are given in terms of the quantities: H = B~ · ~v d3x (21) CNB Z t −1 ∂χ α ≡ (α, β, σ), α [χ ,ν]= −A ( j + ∇~ ν · ∇~ χ ). in which the topological velocity field is defined as fol- i i i ij ∂t j (11) lows: And the generalized Clebsch representation of the veloc- ~v = ~v − σ∇~ S (22) ity [9]: t

It should be noticed that HCNB is conserved even for an ~v = ∇~ ν + α∇~ χ + β∇~ η + σ∇~ S. (12) MHD not satisfying the Sakurai topological constraint 3 given in equation (8), provided that we have a field σ barotropic cross helicity. To conclude we introduce also dσ satisfying the equation dt = T . Thus the non barotropic a local topological conservation law in the spirit of [19] cross helicity conservation law: which is the non barotropic cross helicity per unit of magnetic flux. This quantity which is equal to the disconti- dH CNB =0, (23) nuity of ν is conserved and can be written as a sum of dt the barotropic cross helicity per unit flux and the closed line integral of Sdσ along a magnetic field line: is more general than the variational principle described by equation (9) as follows from a direct computation dH dH using equations (2,4,5,6). Also notice that for a con- [ν]= CNB = C + Sdσ. (24) dΦ dΦ I stant specific entropy S we obtain HCNB = HC and the non-barotropic cross helicity reduces to the standard

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